20:
777:
be made in a short enough interval that it can represent the instantaneous value of the signal with the highest frequency. This means that in the FM radio example above, the sampling circuit must be able to capture a signal with a frequency of 108 MHz, not 43.2 MHz. Thus, the sampling frequency may be only a little bit greater than 43.2 MHz, but the input bandwidth of the system must be at least 108 MHz. Similarly, the accuracy of the sampling timing, or
413:
401:
29:
1129:
776:
When undersampling a real-world signal, the sampling circuit must be fast enough to capture the highest signal frequency of interest. Theoretically, each sample should be taken during an infinitesimally short interval, but this is not practically feasible. Instead, the sampling of the signal should
23:
Fig 1: The top 2 graphs depict
Fourier transforms of 2 different functions that produce the same results when sampled at a particular rate. The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband. The lower
1287:
On the other hand, reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis, recognizing the spectrum mirroring when
32:
Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to undersampling with the maximum value of
1279:
668:
115:. When the aliases are mutually exclusive (spectrally), the original transform and the original continuous function, or a frequency-shifted version of it (if desired), can be recovered from the samples. The first and third graphs of Figure 1 depict a
907:
978:
569:
355:
756:
771:= 4, the FM band spectrum fits easily between 1.5 and 2.0 times the sampling rate, for a sampling rate near 56 MHz (multiples of the Nyquist frequency being 28, 56, 84, 112, etc.). See the illustrations at the right.
275:
958:
1295:
Further generalizations of undersampling for the case of signals with multiple bands are possible, and signals over multidimensional domains (space or space-time) and have been worked out in detail by
1176:
134:) (shaded blue) and its mirror image (shaded beige). The condition for a non-destructive sample rate is that the aliases of both bands do not overlap when shifted by all integer multiples of
582:
141:. The fourth graph depicts the spectral result of sampling at the same rate as the baseband function. The rate was chosen by finding the lowest rate that is an integer sub-multiple of
160:. Consequently, the bandpass function has effectively been converted to baseband. All the other rates that avoid overlap are given by these more general criteria, where
1124:{\displaystyle \left(-{\frac {n}{2}}f_{\mathrm {s} },-{\frac {n-1}{2}}f_{\mathrm {s} }\right)\cup \left({\frac {n-1}{2}}f_{\mathrm {s} },{\frac {n}{2}}f_{\mathrm {s} }\right)}
830:
758:
and this is a scenario of undersampling. In this case, the signal spectrum fits between 2 and 2.5 times the sampling rate (higher than 86.4–88 MHz but lower than 108–110 MHz).
408:= 5) sampling. An anti-alias filter quite tight to the FM radio band is required, and there's not room for stations at nearby expansion channels such as 87.9 without aliasing.
1420:
1313:
1150:
468:
790:
If the sampling theorem is interpreted as requiring twice the highest frequency, then the required sampling rate would be assumed to be greater than the
284:
420:= 4) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (even
80:
of the high-frequency signal. Such sampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF-to-digital conversion.
692:
1413:
1344:
195:
1458:
914:
1406:
1448:
1371:
397:). For the case of a given sampling frequency, simpler formulae for the constraints on the signal's spectral band are given below.
1622:
1603:
1544:
101:
367:
Important signals of this sort include a radio's intermediate-frequency (IF), radio-frequency (RF) signal, and the individual
1274:{\displaystyle n\operatorname {sinc} \left({\frac {nt}{T}}\right)-(n-1)\operatorname {sinc} \left({\frac {(n-1)t}{T}}\right)}
1361:
1489:
663:{\displaystyle 1\leq n\leq \lfloor 5.4\rfloor =\left\lfloor {108\ \mathrm {MHz} \over 20\ \mathrm {MHz} }\right\rfloor }
1585:
54:
782:
1565:
1165:
The corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses
1658:
1538:
1515:
1308:
1532:
1429:
1479:
76:
When one undersamples a bandpass signal, the samples are indistinguishable from the samples of a low-frequency
1474:
1388:
778:
902:{\displaystyle \scriptstyle \left(-{\frac {1}{2}}f_{\mathrm {s} },{\frac {1}{2}}f_{\mathrm {s} }\right),}
122:
The second graph of Figure 1 depicts the frequency profile of a bandpass function occupying the band (
1598:
1527:
1510:
808:
794:
216 MHz. While this does satisfy the last condition on the sampling rate, it is grossly oversampled.
104:) is still available. The individual frequency-shifted copies of the original transform are called
24:
graphs indicate how identical spectral results are created by the aliases of the sampling process.
1555:
1550:
1522:
97:
1612:
1570:
1560:
1484:
1453:
1367:
1340:
89:
42:
19:
1334:
1443:
1296:
911:
and the reconstructive interpolation function, or lowpass filter impulse response, is
804:
70:
785:, must be appropriate for the frequencies being sampled 108MHz, not the lower sample rate.
1134:
119:
spectrum before and after being sampled at a rate that completely separates the aliases.
1159:= 1 (except that where the intervals come together at 0 frequency, they can be closed).
564:{\displaystyle W=f_{H}-f_{L}=108\ \mathrm {MHz} -88\ \mathrm {MHz} =20\ \mathrm {MHz} }
1652:
1627:
1575:
412:
400:
1617:
1608:
350:{\displaystyle 1\leq n\leq \left\lfloor {\frac {f_{H}}{f_{H}-f_{L}}}\right\rfloor }
146:
66:
364:
for which the condition is satisfied leads to the lowest possible sampling rates.
108:. The frequency offset between adjacent aliases is the sampling-rate, denoted by
1505:
416:
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (
404:
Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (
372:
62:
815:
As we have seen, the normal baseband condition for reversible sampling is that
1637:
751:{\displaystyle 43.2\ \mathrm {MHz} <f_{\mathrm {s} }<44\ \mathrm {MHz} }
1398:
449:
1593:
433:
116:
77:
58:
28:
382:> 1, then the conditions result in what is sometimes referred to as
270:{\displaystyle {\frac {2f_{H}}{n}}\leq f_{s}\leq {\frac {2f_{L}}{n-1}}}
970:) = 0 outside the union of open positive and negative frequency bands
953:{\displaystyle \scriptstyle \operatorname {sinc} \left(t/T\right).}
411:
399:
93:
27:
18:
1402:
767:
will also lead to a useful sampling rate. For example, using
962:
To accommodate undersampling, the bandpass condition is that
441:
In the US, FM radio operates on the frequency band from
390:, or using a sampling rate less than the Nyquist rate (2
1363:
Simulation and
Software Radio for Mobile Communications
918:
834:
92:
of real-valued functions are symmetrical around the 0
1179:
1155:
which includes the normal baseband condition as case
1137:
981:
917:
833:
695:
585:
471:
287:
198:
1584:
1498:
1467:
1436:
689:= 5 gives the lowest sampling frequencies interval
1273:
1144:
1123:
952:
901:
750:
662:
563:
349:
269:
73:), but is still able to reconstruct the signal.
1314:Oversampling and undersampling in data analysis
1414:
8:
604:
598:
1421:
1407:
1399:
577:The sampling conditions are satisfied for
1240:
1193:
1178:
1141:
1136:
1109:
1108:
1094:
1084:
1083:
1061:
1041:
1040:
1018:
1005:
1004:
990:
980:
933:
916:
883:
882:
868:
858:
857:
843:
832:
737:
721:
720:
702:
694:
642:
623:
614:
584:
550:
530:
510:
495:
482:
470:
334:
321:
310:
304:
286:
247:
237:
228:
209:
199:
197:
436:to illustrate the idea of undersampling.
1325:
1360:Hiroshi Harada, Ramjee Prasad (2002).
1336:Mixed-signal and DSP design techniques
459:= 108 MHz. The bandwidth is given by
7:
799:Note that if a band is sampled with
1110:
1085:
1042:
1006:
884:
859:
744:
741:
738:
722:
709:
706:
703:
649:
646:
643:
630:
627:
624:
557:
554:
551:
537:
534:
531:
517:
514:
511:
16:Signal processing sample technique
14:
100:of the Fourier transform (called
37:in the equations of this section.
1459:Nyquist–Shannon sampling theorem
145:and also satisfies the baseband
1545:Discrete-time Fourier transform
781:of the sampler, frequently the
102:discrete-time Fourier transform
1255:
1243:
1227:
1215:
811:, instead of a lowpass filter.
96:axis. After sampling, only a
1:
1490:Statistical signal processing
1389:"Undersampling SODAR Signals"
783:analog-to-digital converter
1675:
1539:Discrete Fourier transform
1516:Matched Z-transform method
1309:Drizzle (image processing)
1131:for some positive integer
823:) = 0 outside the interval
1533:Discrete cosine transform
1430:Digital signal processing
53:is a technique where one
1566:Post's inversion formula
1480:Digital image processing
680:can be 1, 2, 3, 4, or 5.
1475:Audio signal processing
1339:. Newnes. p. 20.
1275:
1146:
1125:
954:
903:
752:
664:
565:
425:
409:
351:
271:
61:-filtered signal at a
38:
25:
1276:
1147:
1126:
955:
904:
753:
665:
566:
415:
403:
352:
272:
31:
22:
1599:Anti-aliasing filter
1528:Constant-Q transform
1511:Advanced z-transform
1333:Walt Kester (2003).
1177:
1135:
979:
915:
831:
809:anti-aliasing filter
807:is required for the
779:aperture uncertainty
693:
583:
469:
285:
196:
1145:{\displaystyle n\,}
1556:Integral transform
1551:Impulse invariance
1523:Bilinear transform
1271:
1142:
1121:
950:
949:
899:
898:
748:
660:
561:
426:
410:
347:
277:, for any integer
267:
98:periodic summation
90:Fourier transforms
39:
26:
1659:Signal processing
1646:
1645:
1571:Starred transform
1561:Laplace transform
1485:Speech processing
1454:Estimation theory
1346:978-0-7506-7611-3
1265:
1206:
1102:
1077:
1034:
998:
876:
851:
763:A lower value of
736:
701:
654:
641:
622:
549:
529:
509:
388:bandpass sampling
341:
265:
219:
156: > 2
147:Nyquist criterion
69:(twice the upper
51:bandpass sampling
43:signal processing
1666:
1444:Detection theory
1423:
1416:
1409:
1400:
1393:
1392:
1387:Angelo Ricotta.
1384:
1378:
1377:
1366:. Artech House.
1357:
1351:
1350:
1330:
1280:
1278:
1277:
1272:
1270:
1266:
1261:
1241:
1211:
1207:
1202:
1194:
1151:
1149:
1148:
1143:
1130:
1128:
1127:
1122:
1120:
1116:
1115:
1114:
1113:
1103:
1095:
1090:
1089:
1088:
1078:
1073:
1062:
1052:
1048:
1047:
1046:
1045:
1035:
1030:
1019:
1011:
1010:
1009:
999:
991:
959:
957:
956:
951:
945:
941:
937:
908:
906:
905:
900:
894:
890:
889:
888:
887:
877:
869:
864:
863:
862:
852:
844:
805:band-pass filter
757:
755:
754:
749:
747:
734:
727:
726:
725:
712:
699:
669:
667:
666:
661:
659:
655:
653:
652:
639:
634:
633:
620:
615:
570:
568:
567:
562:
560:
547:
540:
527:
520:
507:
500:
499:
487:
486:
356:
354:
353:
348:
346:
342:
340:
339:
338:
326:
325:
315:
314:
305:
276:
274:
273:
268:
266:
264:
253:
252:
251:
238:
233:
232:
220:
215:
214:
213:
200:
172:are replaced by
71:cutoff frequency
1674:
1673:
1669:
1668:
1667:
1665:
1664:
1663:
1649:
1648:
1647:
1642:
1580:
1494:
1463:
1449:Discrete signal
1432:
1427:
1397:
1396:
1386:
1385:
1381:
1374:
1359:
1358:
1354:
1347:
1332:
1331:
1327:
1322:
1305:
1242:
1236:
1195:
1189:
1175:
1174:
1133:
1132:
1104:
1079:
1063:
1060:
1056:
1036:
1020:
1000:
986:
982:
977:
976:
929:
925:
913:
912:
878:
853:
839:
835:
829:
828:
803:> 1, then a
716:
691:
690:
635:
616:
610:
581:
580:
491:
478:
467:
466:
457:
446:
395:
330:
317:
316:
306:
300:
283:
282:
254:
243:
239:
224:
205:
201:
194:
193:
184:
177:
154:
139:
113:
86:
17:
12:
11:
5:
1672:
1670:
1662:
1661:
1651:
1650:
1644:
1643:
1641:
1640:
1635:
1630:
1625:
1620:
1615:
1606:
1601:
1596:
1590:
1588:
1582:
1581:
1579:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1542:
1536:
1530:
1525:
1520:
1519:
1518:
1513:
1502:
1500:
1496:
1495:
1493:
1492:
1487:
1482:
1477:
1471:
1469:
1465:
1464:
1462:
1461:
1456:
1451:
1446:
1440:
1438:
1434:
1433:
1428:
1426:
1425:
1418:
1411:
1403:
1395:
1394:
1379:
1372:
1352:
1345:
1324:
1323:
1321:
1318:
1317:
1316:
1311:
1304:
1301:
1285:
1284:
1283:
1282:
1269:
1264:
1260:
1257:
1254:
1251:
1248:
1245:
1239:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1210:
1205:
1201:
1198:
1192:
1188:
1185:
1182:
1163:
1162:
1161:
1160:
1153:
1140:
1119:
1112:
1107:
1101:
1098:
1093:
1087:
1082:
1076:
1072:
1069:
1066:
1059:
1055:
1051:
1044:
1039:
1033:
1029:
1026:
1023:
1017:
1014:
1008:
1003:
997:
994:
989:
985:
948:
944:
940:
936:
932:
928:
924:
921:
897:
893:
886:
881:
875:
872:
867:
861:
856:
850:
847:
842:
838:
813:
812:
796:
795:
787:
786:
773:
772:
760:
759:
746:
743:
740:
733:
730:
724:
719:
715:
711:
708:
705:
698:
682:
681:
673:
672:
671:
670:
658:
651:
648:
645:
638:
632:
629:
626:
619:
613:
609:
606:
603:
600:
597:
594:
591:
588:
574:
573:
572:
571:
559:
556:
553:
546:
543:
539:
536:
533:
526:
523:
519:
516:
513:
506:
503:
498:
494:
490:
485:
481:
477:
474:
461:
460:
455:
444:
438:
437:
393:
358:
357:
345:
337:
333:
329:
324:
320:
313:
309:
303:
299:
296:
293:
290:
263:
260:
257:
250:
246:
242:
236:
231:
227:
223:
218:
212:
208:
204:
186:, respectively
182:
175:
152:
137:
111:
85:
82:
15:
13:
10:
9:
6:
4:
3:
2:
1671:
1660:
1657:
1656:
1654:
1639:
1636:
1634:
1633:Undersampling
1631:
1629:
1628:Sampling rate
1626:
1624:
1621:
1619:
1616:
1614:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1591:
1589:
1587:
1583:
1577:
1576:Zak transform
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1529:
1526:
1524:
1521:
1517:
1514:
1512:
1509:
1508:
1507:
1504:
1503:
1501:
1497:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1472:
1470:
1466:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1441:
1439:
1435:
1431:
1424:
1419:
1417:
1412:
1410:
1405:
1404:
1401:
1390:
1383:
1380:
1375:
1373:1-58053-044-3
1369:
1365:
1364:
1356:
1353:
1348:
1342:
1338:
1337:
1329:
1326:
1319:
1315:
1312:
1310:
1307:
1306:
1302:
1300:
1298:
1297:Igor Kluvánek
1293:
1291:
1267:
1262:
1258:
1252:
1249:
1246:
1237:
1233:
1230:
1224:
1221:
1218:
1212:
1208:
1203:
1199:
1196:
1190:
1186:
1183:
1180:
1173:
1172:
1171:
1170:
1169:
1168:
1158:
1154:
1138:
1117:
1105:
1099:
1096:
1091:
1080:
1074:
1070:
1067:
1064:
1057:
1053:
1049:
1037:
1031:
1027:
1024:
1021:
1015:
1012:
1001:
995:
992:
987:
983:
975:
974:
973:
972:
971:
969:
965:
960:
946:
942:
938:
934:
930:
926:
922:
919:
909:
895:
891:
879:
873:
870:
865:
854:
848:
845:
840:
836:
826:
822:
818:
810:
806:
802:
798:
797:
793:
789:
788:
784:
780:
775:
774:
770:
766:
762:
761:
731:
728:
717:
713:
696:
688:
684:
683:
679:
675:
674:
656:
636:
617:
611:
607:
601:
595:
592:
589:
586:
579:
578:
576:
575:
544:
541:
524:
521:
504:
501:
496:
492:
488:
483:
479:
475:
472:
465:
464:
463:
462:
458:
451:
447:
440:
439:
435:
431:
428:
427:
423:
419:
414:
407:
402:
398:
396:
389:
385:
384:undersampling
381:
376:
374:
370:
365:
363:
343:
335:
331:
327:
322:
318:
311:
307:
301:
297:
294:
291:
288:
280:
261:
258:
255:
248:
244:
240:
234:
229:
225:
221:
216:
210:
206:
202:
192:
191:
190:
189:
185:
178:
171:
167:
163:
159:
155:
148:
144:
140:
133:
129:
125:
120:
118:
114:
107:
103:
99:
95:
91:
83:
81:
79:
74:
72:
68:
64:
60:
56:
52:
48:
47:undersampling
44:
36:
30:
21:
1632:
1623:Quantization
1618:Oversampling
1609:Nyquist rate
1604:Downsampling
1382:
1362:
1355:
1335:
1328:
1294:
1289:
1286:
1166:
1164:
1156:
967:
963:
961:
910:
824:
820:
816:
814:
800:
792:Nyquist rate
791:
768:
764:
686:
677:
453:
442:
429:
421:
417:
405:
391:
387:
383:
379:
377:
368:
366:
361:
360:The highest
359:
281:satisfying:
278:
187:
180:
173:
169:
165:
161:
157:
150:
142:
135:
131:
127:
123:
121:
109:
105:
87:
75:
67:Nyquist rate
50:
46:
40:
34:
1506:Z-transform
676:Therefore,
373:filter bank
84:Description
63:sample rate
1638:Upsampling
1499:Techniques
1468:Sub-fields
1320:References
685:The value
65:below its
1613:frequency
1292:is even.
1250:−
1234:
1222:−
1213:−
1187:
1068:−
1054:∪
1025:−
1016:−
988:−
923:
841:−
605:⌋
599:⌊
596:≤
590:≤
522:−
489:−
432:Consider
328:−
298:≤
292:≤
259:−
235:≤
222:≤
1653:Category
1594:Aliasing
1586:Sampling
1303:See also
657:⌋
612:⌊
434:FM radio
430:Example:
369:channels
344:⌋
302:⌊
117:baseband
59:bandpass
827:
106:aliases
55:samples
1547:(DTFT)
1437:Theory
1370:
1343:
735:
700:
640:
621:
548:
528:
508:
1541:(DFT)
1535:(DCT)
448:= 88
371:of a
78:alias
1368:ISBN
1341:ISBN
1231:sinc
1184:sinc
920:sinc
729:<
714:<
697:43.2
452:to
179:and
164:and
88:The
618:108
602:5.4
505:108
450:MHz
378:If
149::
49:or
41:In
1655::
1611:/
1299:.
732:44
637:20
545:20
525:88
424:).
386:,
375:.
126:,
94:Hz
57:a
45:,
1422:e
1415:t
1408:v
1391:.
1376:.
1349:.
1290:n
1281:.
1268:)
1263:T
1259:t
1256:)
1253:1
1247:n
1244:(
1238:(
1228:)
1225:1
1219:n
1216:(
1209:)
1204:T
1200:t
1197:n
1191:(
1181:n
1167::
1157:n
1152:.
1139:n
1118:)
1111:s
1106:f
1100:2
1097:n
1092:,
1086:s
1081:f
1075:2
1071:1
1065:n
1058:(
1050:)
1043:s
1038:f
1032:2
1028:1
1022:n
1013:,
1007:s
1002:f
996:2
993:n
984:(
968:f
966:(
964:X
947:.
943:)
939:T
935:/
931:t
927:(
896:,
892:)
885:s
880:f
874:2
871:1
866:,
860:s
855:f
849:2
846:1
837:(
825::
821:f
819:(
817:X
801:n
769:n
765:n
745:z
742:H
739:M
723:s
718:f
710:z
707:H
704:M
687:n
678:n
650:z
647:H
644:M
631:z
628:H
625:M
608:=
593:n
587:1
558:z
555:H
552:M
542:=
538:z
535:H
532:M
518:z
515:H
512:M
502:=
497:L
493:f
484:H
480:f
476:=
473:W
456:H
454:f
445:L
443:f
422:n
418:n
406:n
394:H
392:f
380:n
362:n
336:L
332:f
323:H
319:f
312:H
308:f
295:n
289:1
279:n
262:1
256:n
249:L
245:f
241:2
230:s
226:f
217:n
211:H
207:f
203:2
188::
183:H
181:f
176:L
174:f
170:B
168:+
166:A
162:A
158:B
153:s
151:f
143:A
138:s
136:f
132:B
130:+
128:A
124:A
112:s
110:f
35:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
